一类非自治随机波动方程的随机吸引子
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摘要
考虑带加性噪声的非自治随机波动方程在R3的有界区域D上的渐近行为.首先将随机偏微分方程转化为仅含随机参数的随机方程,然后运用解的一致估计方法证明随机吸收集的存在性,进一步利用压缩函数方法获得渐近紧性,最后得到随机动力系统拉回吸引子的存在性.
In this paper,we study the asymptotic behavior of non-autonomous stochastic wave equations with additive noise on a bounded domain D R3. Firstly,the partial differential equation is transfromed into the random equation that only includes the random parameters. Then the existence of pullback absorbing set is proved by uniform estimates of solution method. Moreover,from constructing contractive functions,asymptotic compactness is obtained. Finally,the existence of random attractor is given.
In this paper,we study the asymptotic behavior of non-autonomous stochastic wave equations with additive noise on a bounded domain D R3. Firstly,the partial differential equation is transfromed into the random equation that only includes the random parameters. Then the existence of pullback absorbing set is proved by uniform estimates of solution method. Moreover,from constructing contractive functions,asymptotic compactness is obtained. Finally,the existence of random attractor is given.
引文
[1]WANG B. Random attractors for wave equation on unbounded domains[J]. Discrete and Continuous Dynamical System,2009,4(1):800-809.
[2]WANG B. Asymptotic behavior of stochastic wave equations with critical exponents on R3[J]. Transaction of the American Mathematical Society,2011,363(7):3639-3663.
[3]DELL’ORO F. Global attractors for strongly damped wave equations with subcritical-critical nonlinearities[J]. Commun Pure Appl Anal,2013,12(2):1015-1027.
[4]BALL J M. Global attractors for damped semilinear wave equations[J]. Discrete and Continuous Dynamical Systems,2004,10(2):31-52
[5] SUN C Y,CAO D M,DUAN J Q. Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity[J]. Nonlinearity,2006,19(1):2645-2665.
[6]韩英豪,马松.一类非自治随机波动方程拉回吸引子的存在性[J].辽宁师范大学学报(自然科学版),2014,37(4):441-451.
[7]徐明亮.两类具有白噪声和记忆项的波动方程的D-拉回吸引子[D].湘潭:湘潭大学,2016.
[8]WANG B. Random attractors for non-autonomous stochastic wave equations with muliplicative noise[J]. Discrete and Continuous Dynamical System,2014,34(1):269-300.
[9]WANG Z J,ZHOU S F. Random attractor for stochastic non-autonomous damped wave equation with critical exponent[J]. Discrete and Continuous Dynamical System,2017,37(1):545-573.
[10]TEMAM R. Infinite-dimensional System in Mechanics and Physics[M]. New York:Springer-Verlag,1988.
[11]CRAUEL H,FLAUDOLI F. Attractors for random dynamical system[J]. Probability Theory and Related Fields,1994,100:365-393.
[12]CRAUEL H,DEBUSSECHE A,FANCO F. Random attractors[J]. J Dyn Diff Eqns,1997,9(2):307-341.
[13]ARNOLD L. Random Dynamical System[M]. New York:Springer-Verlag,1998.
[14]PETER E K,MARTIN R. Nonautonomous Dynamical System[M]. Providence:American Mathematical Society,2011.
[15]范小明.非自治无穷维动力系统和随机动力系统渐近行为的研究[D].成都:四川大学,2004.
[16]ZHAO C D,ZHOU S F. Random attractor for stochastic reaction-diffusion equation with muliplicative noise on unbouded domains[J]. J Math Anal Appl,2011,384:160-172.
[17]BATES P W,LU K N,WANG B X. Random attractors for stochastic reaction-diffusion equations on unbouded domains[J].J Dyn Diff Eqns,2009,246:845-869.
[18]WANG X H,LI S Y,XU D Y. Random attractors for second-order stochastic lattic dynamical systems[J]. Nonlinear Anal,2010,72(1):483-393.
[19]HUANG J H. The random attractor of stochastic Fitz Hugh-Nagumo equations in infinite lattice with white niose[J]. Physica D:Nonlinear Phnomena,2007,233(2):83-94.
[20]GUO B,WANG G L,LI D L. The attractor of stochastic generalized Ginzburg-Landau equation[J]. Science in China:Mathematics,2008,A51(5):955-964.
[21]王蕊,李扬荣.带有可乘白噪声的广义Ginzburg-Landau方程的随机吸引子[J].西南大学学报(自然科学版),2012,34(2):92-95.
[22]张佳,舒级,董建,等.具乘性噪声的广义Ginzburg-Landau方程的随机吸引子[J].四川师范大学学报(自然科学版),2015,38(5):638-643.
[23]鲍杰,舒级.高阶广义2D Ginzburg-Landau方程的随机吸引子[J].四川师范大学学报(自然科学版),2014,37(3):298-306.
[24]蔡东洪,范小明.具乘性白噪声耗散Kd V型方程的随机吸引子[J].西南民族大学学报(自然科学版),2014,40(4):900-904.
[25]张倩,范小明.耗散MKdV型方程的整体吸引子[J].云南民族大学学报(自然科学版),2012,22(2):128-131.
[26]GARCIA-LUENGO J,MARIN-RUBIO P,REAL J. Pullback attractors in V for non-autonomous 2D-Navier-Stokes equations and their tempered behavior[J]. J Diff Eqns,2012,252(8):4333-4356.
[27]韩英豪,贺亚静. RN上具有记忆项的非自治反应扩散方程的拉回吸引子的存在性[J].辽宁师范大学学报(自然科学版),2012,35(4):441-448.
[2]WANG B. Asymptotic behavior of stochastic wave equations with critical exponents on R3[J]. Transaction of the American Mathematical Society,2011,363(7):3639-3663.
[3]DELL’ORO F. Global attractors for strongly damped wave equations with subcritical-critical nonlinearities[J]. Commun Pure Appl Anal,2013,12(2):1015-1027.
[4]BALL J M. Global attractors for damped semilinear wave equations[J]. Discrete and Continuous Dynamical Systems,2004,10(2):31-52
[5] SUN C Y,CAO D M,DUAN J Q. Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity[J]. Nonlinearity,2006,19(1):2645-2665.
[6]韩英豪,马松.一类非自治随机波动方程拉回吸引子的存在性[J].辽宁师范大学学报(自然科学版),2014,37(4):441-451.
[7]徐明亮.两类具有白噪声和记忆项的波动方程的D-拉回吸引子[D].湘潭:湘潭大学,2016.
[8]WANG B. Random attractors for non-autonomous stochastic wave equations with muliplicative noise[J]. Discrete and Continuous Dynamical System,2014,34(1):269-300.
[9]WANG Z J,ZHOU S F. Random attractor for stochastic non-autonomous damped wave equation with critical exponent[J]. Discrete and Continuous Dynamical System,2017,37(1):545-573.
[10]TEMAM R. Infinite-dimensional System in Mechanics and Physics[M]. New York:Springer-Verlag,1988.
[11]CRAUEL H,FLAUDOLI F. Attractors for random dynamical system[J]. Probability Theory and Related Fields,1994,100:365-393.
[12]CRAUEL H,DEBUSSECHE A,FANCO F. Random attractors[J]. J Dyn Diff Eqns,1997,9(2):307-341.
[13]ARNOLD L. Random Dynamical System[M]. New York:Springer-Verlag,1998.
[14]PETER E K,MARTIN R. Nonautonomous Dynamical System[M]. Providence:American Mathematical Society,2011.
[15]范小明.非自治无穷维动力系统和随机动力系统渐近行为的研究[D].成都:四川大学,2004.
[16]ZHAO C D,ZHOU S F. Random attractor for stochastic reaction-diffusion equation with muliplicative noise on unbouded domains[J]. J Math Anal Appl,2011,384:160-172.
[17]BATES P W,LU K N,WANG B X. Random attractors for stochastic reaction-diffusion equations on unbouded domains[J].J Dyn Diff Eqns,2009,246:845-869.
[18]WANG X H,LI S Y,XU D Y. Random attractors for second-order stochastic lattic dynamical systems[J]. Nonlinear Anal,2010,72(1):483-393.
[19]HUANG J H. The random attractor of stochastic Fitz Hugh-Nagumo equations in infinite lattice with white niose[J]. Physica D:Nonlinear Phnomena,2007,233(2):83-94.
[20]GUO B,WANG G L,LI D L. The attractor of stochastic generalized Ginzburg-Landau equation[J]. Science in China:Mathematics,2008,A51(5):955-964.
[21]王蕊,李扬荣.带有可乘白噪声的广义Ginzburg-Landau方程的随机吸引子[J].西南大学学报(自然科学版),2012,34(2):92-95.
[22]张佳,舒级,董建,等.具乘性噪声的广义Ginzburg-Landau方程的随机吸引子[J].四川师范大学学报(自然科学版),2015,38(5):638-643.
[23]鲍杰,舒级.高阶广义2D Ginzburg-Landau方程的随机吸引子[J].四川师范大学学报(自然科学版),2014,37(3):298-306.
[24]蔡东洪,范小明.具乘性白噪声耗散Kd V型方程的随机吸引子[J].西南民族大学学报(自然科学版),2014,40(4):900-904.
[25]张倩,范小明.耗散MKdV型方程的整体吸引子[J].云南民族大学学报(自然科学版),2012,22(2):128-131.
[26]GARCIA-LUENGO J,MARIN-RUBIO P,REAL J. Pullback attractors in V for non-autonomous 2D-Navier-Stokes equations and their tempered behavior[J]. J Diff Eqns,2012,252(8):4333-4356.
[27]韩英豪,贺亚静. RN上具有记忆项的非自治反应扩散方程的拉回吸引子的存在性[J].辽宁师范大学学报(自然科学版),2012,35(4):441-448.